Abstract
Mediation analysis often requires larger sample sizes than main effect analysis to achieve the same statistical power. Combining results across similar trials may be the only practical option for increasing statistical power for mediation analysis in some situations. In this paper, we propose a method to estimate: (1) marginal means for mediation path a, the relation of the independent variable to the mediator; (2) marginal means for path b, the relation of the mediator to the outcome, across multiple trials; and (3) the between-trial level variance–covariance matrix based on a bivariate normal distribution. We present the statistical theory and an R computer program to combine regression coefficients from multiple trials to estimate a combined mediated effect and confidence interval under a random effects model. Values of coefficients a and b, along with their standard errors from each trial are the input for the method. This marginal likelihood based approach with Monte Carlo confidence intervals provides more accurate inference than the standard meta-analytic approach. We discuss computational issues, apply the method to two real-data examples and make recommendations for the use of the method in different settings.
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References
Arbuckle JL (1995) Amos 7.0 user’s guide. Amos Development Corporation
Becker BJ, Schram CM (1994) Examining explanatory models through research synthesis. In: Portions of this chapter were presented at the annual meeting of the American Educational Research Association in San Francisco, CA, 27 Mar 1989. Russell Sage Foundation
Brown CH, Chamberlain P, Saldana L, Padgett C, Wang W, Cruden G (2014) Evaluation of two implementation strategies in 51 child county public service systems in two states: results of a cluster randomized head-to-head implementation trial. Implement Sci 9(1):134
Brown CH, Sloboda Z, Faggiano F, Teasdale B, Keller F, Burkhart G, Vigna-Taglianti F, Howe G, Masyn K, Wang W et al (2013) Methods for synthesizing findings on moderation effects across multiple randomized trials. Prev Sci 14(2):144–156
Byrd RH, Lu P, Nocedal J, Zhu C (1995) A limited memory algorithm for bound constrained optimization. SIAM J Sci Comput 16(5):1190–1208
Curran PJ, Hussong AM (2009) Integrative data analysis: the simultaneous analysis of multiple data sets. Psychol Methods 14(2):81
Fritz MS, MacKinnon DP (2007) Required sample size to detect the mediated effect. Psychol Sci 18(3):233–239
Harris MJ, Rosenthal R (1985) Mediation of interpersonal expectancy effects: 31 meta-analyses. Psychol Bull 97(3):363
Horowitz JL (2001) The bootstrap. Handb Econom 5:3159–3228
Insel TR, Gogtay N (2014) National institute of mental health clinical trials: new opportunities, new expectations. JAMA Psychiatry 71(7):745–746
MacKinnon DP (2008) Introduction to statistical mediation analysis. Routledge, London
MacKinnon DP, Lockwood CM, Williams J (2004) Confidence limits for the indirect effect: distribution of the product and resampling methods. Multivar Behav Res 39(1):99–128
MacKinnon DP, Wurpts IC, Valente MJ (2014) Imagery and memory theory as known effect validation of mediation analysis. Unpublished manuscript
Muthén LK, Muthén BO (2010) Mplus: statistical analysis with latent variables: user’s guide. Muthén & Muthén
Perrino T, Pantin H, Huang S, Brincks A, Brown CH, Prado G (2015) Reducing the risk of internalizing symptoms among high-risk hispanic youth through a family intervention: a randomized controlled trial. Fam Process. doi:10.1111/famp.12132
Perrino T, Pantin H, Prado G, Huang S, Brincks A, Howe G, Beardslee W, Sandler I, Brown CH (2014) Preventing internalizing symptoms among hispanic adolescents: a synthesis across familias unidas trials. Prev Sci 15(6):917–928
Prado G, Pantin H (2011) Reducing substance use and hiv health disparities among hispanic youth in the USA: the familias unidas program of research. Psychosoc Interv 20(1):63–73
Preacher KJ, Hayes AF (2008) Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models. Behav Res Methods 40(3):879–891
Shadish WR (1996) Meta-analysis and the exploration of causal mediating processes: a primer of examples, methods, and issues. Psychol Methods 1(1):47
Springer MD (1979) The algebra of random variables. Wiley, New York
Tofighi D, MacKinnon DP (2011) Rmediation: an R package for mediation analysis confidence intervals. Behav Res Methods 43(3):692–700
Acknowledgments
We are grateful for support from the National Institute on Drug Abuse (NIDA) (P30DA027828, C Hendricks Brown PI, R01DA03399103S1, Cady Berkel PI, and R01DA009757 and R37DA009757, David P. MacKinnon PI), and the National Institute of Mental Health (R01MH040859, C Hendricks Brown, PI). The content of this paper is solely the responsibility of the authors and does not necessarily represent the official views of the funding agencies.
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Appendix: Restricted maximum likelihood estimates for \(\Sigma \)
Appendix: Restricted maximum likelihood estimates for \(\Sigma \)
The differential \(d (\mathbf{V}_i^{-1})\) of this symmetric matrix can be expressed in terms of the the differential of the matrix itself by noting that by the chain rule,
and using \(d ( \mathbf{V}_i^{-1} \mathbf{V}_i ) = d ( I_K ) = 0 \), we have
To find the differential \(d log | \mathbf{V}_i |\) we note that a determinant of any positive definite symmetric matrix \(\varPsi \) can be related to an integral for a multivariate normal distribution.
Thus
where the definite multivariate integral
Differentiating the logarithm on each side of Eq. 24,
Differentiating the integral can be done by bringing the differential inside the integrand of Equation and using the chain rule, so
Equation 27 comes from the definition of the expectation of the random matrix \(\mathbf{U}\mathbf{U}'\), and this expectation evaluates to the variance-covariance matrix for a multivariate normal distribution as in Eq. 28. The final equation in this series comes from Eq. 22. Substituting Eqs. 22 and 29 into Eq. 9 we finally obtain,
Since \(\mathbf{V}_i = diag ( \varPsi _i , \varPhi _i ) + \Sigma , i = 1 , \ldots , K\), we have
Substituting this expression into the previous differential results in three scalar equations, one for each component of the variance-covariance matrix.
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Huang, S., MacKinnon, D.P., Perrino, T. et al. A statistical method for synthesizing mediation analyses using the product of coefficient approach across multiple trials. Stat Methods Appl 25, 565–579 (2016). https://doi.org/10.1007/s10260-016-0354-y
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DOI: https://doi.org/10.1007/s10260-016-0354-y